Annuities Due Chapter 13 McGraw-Hill Ryerson©

Learning Objectives Annuities Due Annuities Due Calculate After completing this chapter, you will be able to: Calculate LO 1. … the Future Value and Present Value of Annuities Due LO 2. …the payment size, number of payments, and interest rate in Annuities Due

Membership dues are usually paid in advance Annuities Due 1st. If you lease equipment, a vehicle, or rent property, the typical lease contract requires payments at the beginning of each period of coverage 15th. Membership dues are usually paid in advance Only a small modification to the math of Ordinary Annuities is needed to handle Annuities Due

are made at the beginning of the period Classification of Annuities Ordinary Annuity Annuitiy Due Payments / Deposits are made at the end of the period Payments / Deposits are made at the beginning of the period

Compare the payment interval to the compounding interval 13 Annuities Due Classification of Annuities 13 - 6 Is the payment at the end or at the beginning of each payment interval? Compare the payment interval to the compounding interval Annuity Category End Equal Ordinary Simple End Not Equal Ordinary General Simple Annuity Due Beginning Equal General Annuity Due Beginning Not Equal McGraw-Hill Ryerson©

[ ] [ ] * (1+ i) = PMT * (1+ i) = PMT Annuity Due i i Formulae FVdue LO 1. FVdue = PMT (1+ i)n - 1 [ i ] Formula * (1+ i) PVdue = PMT 1-(1+ i)-n [ i ] Formula * (1+ i) If the payments form a general annuity due, use i2 = (1+i)c – 1 in the formula for i

Clues to help identify annuities due Tips “Payments at the beginning of each…..” 1. 2. “Payments…in advance” 3. “First payment … made today” “Payments ….. starting now” 4.

How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday.

Financial Q 2nd BGN 2nd SET 2nd QUIT BGN BGN END Step 2… Step 1 – Set your calculator to the “BGN mode” How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q Your calculator is now set for annuity due Notice the tiny BGN above the 0 BGN BGN END 2nd BGN SET 2nd QUIT 2nd Step 2…

Elyse’s RRSP value on her 60th birthday Financial Step 2 – Solve for the FV of the annuity due How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q BGN Elyse’s RRSP value on her 60th birthday FV= 247,510.31 2 2nd ON/OFF P/Y 1000 60 PMT N 8 PV ENTER CPT FV QUIT 2nd Algebraic Calculation

Elyse’s RRSP value on her 60th birthday [ FV = PMT (1+ i)n - 1 i ] * (1+ i) Formula c = 1 i = 0.08/2 PMT = 1000 n = 60 How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q Elyse’s RRSP value on her 60th birthday 247,510.31 10.5196 9.5196 60 1.04 y x - 1 . 0.04 X 1000 X 1.04

A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? What do we have to find? Need to find the PV today of the payment alternative and compare with the $300,000 cash.

Your calculator is now set for annuity due Financial Step 1 – Set your calculator to the “BGN mode” A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? Q Your calculator is now set for annuity due BGN END BGN 2nd BGN SET 2nd QUIT 2nd Step 2…

Cash Value of the payment option Financial Step 2 – Solve for the PV of the annuity due A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? Q BGN Cash Value of the payment option PV= 283,775.83 4 2nd ON/OFF P/Y 7000 80 PMT N 8 FV CPT PV ENTER The $300,000 cash is a better offer …$16,224 more in current day dollars…($300,000 - $283,775.83) QUIT 2nd Algebraic Calculation

[ ] [ ] * (1+ i) = PMT Q i 1-(1+ i)-n PVdue *(1.02) = $283,775.83 A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly? PVdue = PMT 1-(1+ i)-n [ i ] * (1+ i) (1 - (1 + .08/4)-80) .02 *(1.02) = 7000 [ ] = $283,775.83 The $300,000 cash is a better offer… $16,224 more in current day dollars…($300,000 - $283,775.83)

Step 1 – Set your calculator to the “BGN mode” Financial LO 2. Step 1 – Set your calculator to the “BGN mode” Q Step 2 – Solve for the PMT of the annuity due You have accumulated $104,000 in your RRSP. Your goal is to build it to $250,000 with equal contributions every 6 months for the next 7 years. If you earn 8.5% compounded semi-annually, and start today, find the size of your contributions. Step 1 Step 2 PMT= -3,286.10 BGN BGN 2nd BGN 2 2nd QUIT ENTER P/Y 14 8.5 SET 2nd N 250000 FV QUIT 2nd 104000 PV Payment needed is $3286.10 CPT PMT Algebraic Calculation

[ ] Step Q *(1+ i) = PMT - 1 i Extract necessary data... c = 1 i = Formula FVdue = PMT [ (1+ i)n - 1 i ] *(1+ i) Step Extract necessary data... c = 1 i = 0.085/2 PMT = ? n = 14 You have accumulated $104,000 in your RRSP. Your goal is to build it to $250,000 with equal contributions every 6 months for the next 7 years. If you earn 8.5% compounded semi-annually, and start today, find the size of your contributions. Q FV = 250 000 PV = 104 000 Find the FV 7 years from now of the $104,000 already saved. a. FV = 104 000(1+.085/2)14 = $186,250.84 Subtract this value from the $250,000 target to get the FV of the annuity. b. $250,000 - $186,250.84 = $63,749.16

[ ] ] [ Step Choose appropriate formula and Solve FV = PMT (1+ i)n - 1 ? 0.085/2 PMT = 14 1 FV = $63,749.16 PV = 0 (1.0425)14 – 1 0.0425 * (1.0425) 63749.16 = PMT ] [ 63749.16 = PMT(19.3997) PMT = $3286.10

Q A car sells for $27,900. The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is $14,500, find the lease payments assuming a $2,500 down payment. Purchase Price Down Payment Present Value of Lease Payments Present Value of Residual Value = + $27,900 $2,500 Present Value of Lease Payments Present Value of $14,500 = + $25,400 Present Value of Lease Payments = + Present Value of $14,500

Step 1 – Set your calculator to the “BGN mode” Financial Step 1 – Set your calculator to the “BGN mode” Step 2 – Solve for the PMT of the annuity due Step 1 Step 2 A car sells for $27,900. The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is $14,500, find the lease payments assuming a $2,500 down payment. Q BGN PMT= 332.50 BGN 2nd BGN 12 2nd QUIT ENTER P/Y 36 1.8 SET 2nd N 14500 FV QUIT 2nd 25400 PV Payment needed is $322.50 CPT PMT Algebraic Calculation

Find the PV of the $14,500 residual value Step Extract necessary data... c = 1 i = 0.018/12 PMT = ? n = 36 A car sells for $27,900. The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is $14,500, find the lease payments assuming a $2,500 down payment. Q a. Find the PV of the $14,500 residual value PV = 14 500(1+.018/12)-36 = $13,738.32 b. PV annuity = SP – DP – PV of residual value = $27,900 - $2500 - $13,738.32 = $11,661.68 See Next

[ ] [ ] * (1+ i) = PMT Step i 1-(1+ i)-n PVdue Choose appropriate formula and Solve Step PVdue = PMT 1-(1+ i)-n [ i ] * (1+ i) c = 1 i = 0.018/12 PMT = ? n = 36 FV = PV = $11,661.68 [ ] 11661.68 = 1 - (1.0015)-36 0.0015 * (1.0015) PMT 11661.68 = PMT (35.0722) PMT = $332.50 Lease Payment

Q i = .06/4 QUIT 2nd BGN SET 2nd P/Y QUIT ENTER I/Y PV PMT FV CPT N How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP earns 6% compounded quarterly. i = .06/4 PV = $0 n =? FV = $1 000 000 Step 1 Step 2 143 Quarterly payments N = 142.9 BGN BGN QUIT 2nd BGN SET 2nd P/Y 4 QUIT ENTER 35.75 years 6 I/Y PV 2000 PMT 1 000 000 FV CPT N Algebraic Calculation

…using the algebraic method to determine Number of Payments Calculation …using the algebraic method to determine Number of Payments PMT FVdue i * ( ) + 1 ln [ ] n (1+ i) Formulae PMT i * ( ) + 1 ln [ ] n PVdue (1+ i)

How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP earns 6% compounded quarterly. Q n =? PV = $0 FV = $1000000 i = .06/4 PMT FVdue i * ( ) + 1 ln [ ] n = (1+ i) Formula ] [ 1,000,000 * .015 ln (1.015) 2000(1.015) 1+ n = ln = 2.1269 0.0149 = 142.9 i.e. 143 quarterly payments or 35.75 years

Effective Interest rate on the monthly payment plan = 13.04% Q A $100,000 life insurance policy requires an annual premium of $420 or a monthly premium of $37. In either case, the premium is due at the beginning of the period of coverage. What is the effective rate of interest charged to those who pay monthly? 2nd P/Y BGN mode Set to P/Y = 12 BGN C/Y = 1 I/Y = 13.04 12 ENTER 12 N 420 PV 37 PMT 1 ENTER QUIT 2nd FV CPT I/Y Effective Interest rate on the monthly payment plan = 13.04%

This completes Chapter 13